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AP Calc BC in a week - Part 5

Introduction

Yay! Last day! Today is:

  • 10 Sequences and Series

Let's start!

This section is long so I'm going to have to split it up quite a bit...

10 Sequences and Series

What is a(n) (Infinite) Sequence?

An infinite sequence is a function which has the natural numbers as the domain (not including 0). It is denoted as \(a_n\).

An example of a sequence is \(a_n=\frac{1}{n}\), which results in the sequence \(\frac{1}{1},\frac{1}{2},\frac{1}{3}\cdots\frac{1}{n}\cdots\). \(\frac{1}{n}\) is often called the nth term of this sequence.

I suppose this is to contrast sequences from the other functions that we deal with, whose domain consist of all the real numbers.

A sequence \(a\) converges to a finite number \(L\) if \(\lim_{n\to\infty}a_n=L\). If the limit does not exist then the sequence diverges.

tl;dr A sequence is a (function) list of numbers, and if the limit ( \(n\to\infty\) ) of its function is \(L\), then it converges to \(L\).

What is a(n) (Infinite) Series?

An infinite series is the sum of an infinite sequence. $$\sum_{k=1}^\infty a_k = a_1 + a_2 + a_3 + \cdots + a_n + \cdots$$ Each element in the sum is a term, with \(a_n\) being the nth term, like in the sequence.

The series \(\sum_{k=1}^\infty a_k\) can also be written as \(\sum a_k\) or \(\sum a_n\).

tl;dr Add all the numbers in a sequence = series

Types of Series

p-series is the series of form $$\sum_{k=1}^\infty \frac{1}{k^p}$$, where \(p\) is a constant.

Harmonic series is the p-series where \(p=1\), ie $$\sum_{k=1}^\infty \frac{1}{k}$$

Geometric series is the sum of a first term \(a\) and terms of a common ratio \(r\) : $$\sum_{k=1}^\infty ar^{k-1}$$

Convergence of a Series

If there exists a finite number \(S\) such that \(\lim_{n\to\infty}\sum_{k=1}^ne a_k = S\), then the inifinite series is said to converge to \(S\).

In this case, $$\sum_{k=1}^\infty = S$$.

tl;dr If you keep adding and it approaches a number, then the series converges to that number

There are a couple examples of proving that series are Divergent/convergent, but... I don't think I can reproduce these with an unseen problem, since these involve finding limits of summations, so... hopefully they can be reasoned through if they do come out.

Theorems of Convergence of Divergence

If \(\sum a_n\) converges, then \(\lim_{n\to\infty}a_n=0\)

  • By contraposition, this also means that if \(\lim_{k\to\infty}a_n\neq 0\), \(\sum a_n\) diverges
  • The converse is not true; if the limit is zero, the summation may or may not converge (if you know your truth tables this won't be a problem)
  • Note that this means that the terms go to 0, not the series

You can add a finite number of terms to a summation without affecting its convergence/divergence

  • This means that \(\sum_{k=1}^\infty a_k\) and \(\sum_{k=m}^\infty a_k\) both converge or both diverge

A series can be multiplied by a nonzero constant without affecting its convergence/divergence

  • This means that \(\sum_{k=1}^\infty a_k\) and \(c\sum_{k=1}^\infty a_k\) both converge or both diverge

If two series converge, the sum of the two series converges

  • If \(\sum a_n\) and \(\sum b_n\) both converge, \(\sum (a_n+b_n)\) converges

If the terms of a convergent series are regrouped, the new series is also convergent

Conclusion

Uhh okay I've had this buffer open for too long I need to publish this prematurely, plus the post will become way too long anyway.

Chapter 10 is longer than I expected... I'll have to break it up. Next time we'll be doing Tests for convergences, then after that we'll do the Power series.


2016-03-29 Paul Elder

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